I do not want to ask "are animals saved", because animals do not sin, so its not about salvation.
But are animals (well, I am asking specifically about our pets, not about some insects living in the wild) just temporal beings or do they have soul/spirit that lives after their bodies fail like ours? Will we meet them again?
I know that Bible (sadly) does not say much about this. There are few verses that are about animals (like "who knows, whether the spirit of the beast goes down" or Paul's restoration of creation), but these are quite ambiguous. So, what is your opinion or reasoning?
You mentioned an important point in regards to sin and salvation.
We need to understand a few details about Genesis first. Genesis informs us that God breathed into Adam and he became a living Nephesh (soul), which implies that Adam had a body but there was the absence of being (the Self) until God breathed consciousness of self. Eve was created from the side of Adam, though scripture is silent on it, one would assume that God also would have breathed into Eve and she would have become a living soul. The human family is created in the image of God and they fell away requiring salvation through a saviour Jesus Christ and his salvation Cross.
Nothing goes to heaven that does not go through Jesus Christ and nothing lives after biological death that does not go through the redemptive works of the Cross in restoring them to God the Father. In essence in reply to your important point in regards to sin and salvation for animals, they are not accounted for and this comes down to why they were created before Eve in the first place. Animals were created for the purpose of companionship and when Adam fell, the animals also suffered without any wrongdoing of their own and God had not placed any impositions on the animals and in this regard sin and salvation is not applicable to animals and if it is not applicable then to live after biological death requires Jesus Christ and in this regard they cannot be accounted for in the absence of the Cross as the only means by which life after death was granted, in that it was granted to humanity only and through Jesus Christ. Animals do not have this access to heaven and do not have access to the Cross and do not have access to Jesus Christ and in this regard are not accounted for in the transition from this temporal life to the afterlife in order to be with Jesus, let alone those to whom those pets belong to.
My personal opinion of what animals and/or pets are is beside the point, but in answering your questions below, I find it important to relate it to the above-explained -
Do they have soul/spirit that lives after their bodies fail like ours?
If they did then Adam's free will actions owing to a self-aware being should not have also condemned the animals along with him, as there were no impositions made by God upon them. Their bodies biologically die, but there is no free will agency owing to a self-being to continue afterwards.
Will we meet them again?
When God permits you to meet anyone in Heaven, they had to have been availed the Cross and come through the saviour Jesus Christ and as it stands and by your important opening statement in regards to sin and salvation, animals/pets are not accounted for, that is the path you go to be with Jesus is not the path availed to them and where they will be is not where you will be.
So, what is your opinion or reasoning?
This is an open-ended question, so I will opine. But firstly, let us study the verses you pointed to -
Ecclesiastics 3:19-22
For
that which befalleth the sons of men befalleth beasts; even one thing befalleth them: as the one dieth, so dieth the other; yea, they have all one breath; so that a
man hath no preeminence above a beast: for all
is vanity.
20All go unto one place; all are of the dust, and all turn to dust again.
21Who knoweth the spirit of man that goeth upward, and the spirit of the beast that goeth downward to the earth? 22Wherefore I perceive that
there is nothing better, than that
a man should rejoice in his own works; for that is his portion: for who shall bring him to see what shall be after him?
Notice that the author is posing a question and answering it himself.
The question being does the spirit of man go up and the spirit of the beast stays downward in the dust of the earth?
There is nothing better for a man to rejoice at the prospects that his works through the redemptive works of Christ on the Cross shall bring him to realise what shall be after he biologically dies, in contrast to the animals who return to the dust like humans, but with the exceptions of man's works of faith that promises his spirit to rise up. So the writer is answering his own question by saying that though man and animal return to dust, man has the promise to live beyond the grave where the beast will stay down in the earth. The contrasting comparison is that the beast/animal is not going anywhere after they die in the earth, whereas a man may go up if they continue the works of faith in anticipation of what shall be after they biologically die.